Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

Gracar, Peter; Grauer, Arne; Mörters, Peter

doi:10.1007/s00220-022-04445-3

Cited by 6 publications

(3 citation statements)

References 33 publications

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“…We conjecture that if certain lower-order moments below a (model-dependent) critical threshold are infinite, this implies ultra-small distances. This is also the behavior that the authors of [17] observe, for a special class of models, but we believe this behavior is universal.…”

Section: Consequences and Discussionsupporting

confidence: 80%

“…Theorem 3 poses the following question: when are the typical distances exactly doubly logarithmic? Interestingly, distances can be larger than doubly logarithmic, even when the limiting degree distribution has infinite second moment, , as was shown in [17]; see for example [17, Theorem 1.1(a)]. We conjecture that if certain lower-order moments below a (model-dependent) critical threshold are infinite, this implies ultra-small distances.…”

Section: Consequences and Discussionsupporting

confidence: 56%

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Local limits of spatial inhomogeneous random graphs

2023

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Consider a set of n vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$ , and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations and the vertex weights. Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are independent and identically distributed copies of limiting vertex weights. Our set-up covers many sparse geometric random graph models from the literature, including geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs, continuum scale-free percolation, and weight-dependent random connection models. We prove that the limiting degree distribution is mixed Poisson and the typical degree sequence is uniformly integrable, and we obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a byproduct of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

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Section: Consequences and Discussionsupporting

confidence: 80%

Section: Consequences and Discussionsupporting

confidence: 56%

Local limits of spatial inhomogeneous random graphs

2023

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“…The technique of nets combined with multi-round edge-exposure is robust, and will be applicable elsewhere, for questions concerning first passage percolation, robustness to percolation (random deletion of edges), graph distances, SIR-type and other epidemic processes, rumour spreading, etc. on a larger class of vertex-weighted graphs; including random geometric graphs, Boolean models with random radii, the age-dependent and the weight-dependent random connection model (mimicking spatial preferential attachment), scale-free Gilbert graph, and the models used here [2,14,[23][24][25][26][27]32,33], and can also be extended to dynamical versions of the above graph models on fixed vertex sets.…”

Section: -Fpp Parametersmentioning

confidence: 99%

Penalising transmission to hubs in scale-free spatial random graphs

Komjáthy¹,

Lapinskas²,

Lengler³

2021

Ann. Inst. H. Poincaré Probab. Statist.

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In this paper we study a version of (non-Markovian) first passage percolation on graphs, where the transmission time between two connected vertices is non-iid, but increases by a penalty factor polynomial in their expected degrees. Based on the exponent of the penalty-polynomial, this makes it increasingly harder to transmit to and from high-degree vertices. This choice is motivated by awareness or time-limitations. For the iid part of the transmission times we allow any nonnegative distribution with regularly varying behaviour at 0. For the underlying graph models we choose spatial random graphs that have power-law degree distributions, so that the effect of the penalisation becomes visible: (finite and infinite) Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We prove that upon increasing the penalty exponent, the transmission time between two far away vertices x, y sweeps through four universal phases even for a single underlying graph: explosive (tight transmission times), polylogarithmic, polynomial but sublinear (Θ(|x − y| η0 ) for an explicit η 0 < 1), and linear (Θ(|x − y|)) in their Euclidean distance. Further, none of these phases are restricted to phase boundaries, and those are non-trivial in the main model parameters: the tail of the degree-distribution, a long-range parameter, and the exponent of regular variation of the iid part of the transmission times. In this paper we present proofs of lower bounds for the latter two phases and the upper bound for the linear phase. These complement the matching upper bounds for the polynomial regime in our companion paper.

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